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Understanding Symmetric Functions: Even and Odd

Learn about symmetric functions, including even functions with y-axis symmetry and odd functions with origin symmetry. Determine if a function is even, odd, or neither, and explore increasing, decreasing, and constant intervals.

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Understanding Symmetric Functions: Even and Odd

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  1. Symmetric about the y axis FUNCTIONS Symmetric about the origin

  2. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Even functions have y-axis Symmetry 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an even function, for every point (x, y) on the graph, the point (-x, y) is also on the graph. f(-x) = f(x)

  3. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 Odd functions have origin Symmetry 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7 So for an odd function, for every point (x, y) on the graph, the point (-x, -y) is also on the graph. f(-x) = - f(x)

  4. -7 -2 -1 1 3 5 7 -6 -5 -4 -3 0 4 6 8 2 x-axis Symmetry We wouldn’t talk about a function with x-axis symmetry because it wouldn’t BE a function. 8 7 6 5 4 3 2 1 -2 -3 -4 -5 -6 -7

  5. A function is even if f( -x) = f(x) for every number x in the domain. So if you plug a –x into the function and you get the original function back again it is even. Is this function even? YES Is this function even? NO

  6. A function is odd if f( -x) = - f(x) for every number x in the domain. So if you plug a –x into the function and you get the negative of the function back again (all terms change signs) it is odd. Is this function odd? NO Is this function odd? YES

  7. If a function is not even or odd we just say neither (meaning neither even nor odd) Determine if the following functions are even, odd or neither. Not the original and all terms didn’t change signs, so NEITHER. Got f(x) back so EVEN.

  8. Increasing and Decreasing Functions

  9. Increasing/Decreasing Functions A function f is increasing on an interval if as x increases, then f(x) increases. A function f is decreasing on an interval if as x increases, then f(x) decreases. f(x) is decreasing in the interval . f(x) is increasing in the interval . vertex (1.5,-2)

  10. Answer Now Increasing, Decreasing, Constant Intervals A function f is constant on an interval if as xincreases, then f(x) remains the same. Find the interval(s) over which the interval is increasing, decreasing and constant?

  11. Answer Now Increasing, Decreasing, Constant Intervals Find the interval(s) over which the interval is increasing, decreasing and constant?

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